You got one bit wrong
The UK still had our own designed gravity nuclear bombs after going to submarine launched missiles for the strategic deterrent.
But also we bought in the US missiles, but still build our own warheads.
Things did change with Trident though. With Polaris we had UK warheads on a US missile - and in the 70s we spent rather a lot of money on upgrading to Chevaline. It wasn't quite a MIRV system - but was a set of decoys and multiple warheads that re-entered together then split up - so could hit multiple targets at once - but presumably only reasonably close together ones. But that was designed to deal with the Russian ABM defences around Moscow and upgrades to them for the next few years. It was cheaper than designing the whole MIRV thing.
However it was decided that this wouldn't cut the mustard for Trident. So the choice was either to dust off all the Cheveline research and re-do the whole thing, or to ask the US if we could buy their tech and sit our warheads in their existing system. Which was what was agreed - as it was cheaper for us, and they got cash to off-set the reseach costs they'd already spent.